Subgroups — C₂₉×#28823 #20323723

Order 7424.

618 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3164	C₂	1	2	yes	maximal
2	C₂₉×#3163	C₂	1	2	yes	maximal
3	C₂₉×#2061	C₂	1	4	yes	maximal
4	C₂₉×#1952	C₂	1	2	yes	maximal
5	C₂₉×#1951	C₂	1	2	yes	maximal
6	C₂₉×#1846	C₂	1	1	yes	maximal, characteristic
7	C₂₉×#1772	C₂	1	1	yes	maximal, characteristic
8	C₂₉×#1772	C₂	1	1	yes	maximal, characteristic
9	C₂₉×#544	C₂²	1	4	yes	—
10	C₂₉×#501	—	2	4	no	—
11	C₂₉×#501	—	2	4	no	—
12	C₂₉×#500	—	2	2	no	—
13	C₂₉×#499	—	2	2	no	—
14	C₂₉×#499	—	2	2	no	—
15	C₂₉×#498	—	2	2	no	—
16	C₂×C₂₉×#108	C₂²	1	1	yes	characteristic
17	C₂×C₂₉×#108	C₂²	1	1	yes	characteristic
18	C₂₉×#428	C₂²	1	4	yes	—
19	C₂₉×#428	C₂²	1	4	yes	—
20	C₂₉×#427	C₂²	1	1	yes	characteristic
21	C₂₉×#427	C₂²	1	1	yes	characteristic
22	C₂×C₂₉×#107	C₂²	1	2	yes	—
23	C₂×C₂₉×#106	C₂²	1	2	yes	—
24	C₂₉×#405	C₂²	1	2	yes	—
25	C₂₉×#405	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
27	C₂₉×#382	C₂²	1	2	yes	—
28	C₂₉×#343	C₂²	1	4	yes	—
29	C₂₉×#339	C₂²	1	2	yes	—
30	C₂₉×#339	C₂²	1	2	yes	—
31	C₂×M₁₆×C₂₉	C₂³	1	2	yes	—
32	C₂×M₁₆×C₂₉	C₂³	1	2	yes	—
33	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
35	C₂₉×C₄⋊Q₈	—	2	4	no	—
36	C₂₉×C₄⋊Q₈	—	2	4	no	—
37	C₂₉×#125	—	2	4	no	—
38	C₂₉×#125	—	2	4	no	—
39	C₂₉×C₂²⋊Q₈	—	4	4	no	—
40	C₂₉×#117	C₂³	1	2	yes	—
41	C₂₉×#117	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
43	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
44	C₂₉×#108	Q₈	1	4	yes	complemented
45	C₂₉×#108	Q₈	1	4	yes	complemented
46	C₂₉×#107	—	2	2	no	—
47	C₂₉×#107	—	2	2	no	—
48	C₂₉×#107	—	2	4	no	—
49	C₂₉×#107	—	2	4	no	—
50	C₂₉×#107	D₄	1	2	yes	—
51	C₂₉×#107	D₄	1	2	yes	—
52	C₂₉×#106	—	2	2	no	—
53	C₂₉×#106	—	2	2	no	—
54	C₂₉×#106	—	2	4	no	—
55	C₂₉×#106	—	2	4	no	—
56	C₂₉×#106	D₄	1	1	yes	characteristic
57	C₂₉×#106	D₄	1	1	yes	characteristic
58	C₂₉×#106	D₄	1	2	yes	—
59	C₂₉×#97	—	2	2	no	—
60	C₂₉×#97	—	2	2	no	—
61	C₄×C₈×C₂₉	—	2	2	no	—
62	C₄×C₈×C₂₉	—	2	2	no	—
63	C₄×C₈×C₂₉	Q₈	1	4	yes	complemented
64	C₂₉×#95	—	4	2	no	—
65	C₂×Q₈×C₂₉	—	4	4	no	—
66	C₂²×C₄×C₂₉	—	2	1	no	—
67	C₂²×C₄×C₂₉	—	2	4	no	—
68	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₂₉	—	2	2	no	—
70	M₁₆×C₂₉	—	2	2	no	—
71	M₁₆×C₂₉	C₂×Q₈	1	4	yes	—
72	M₁₆×C₂₉	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₂₉	—	2	2	no	—
74	C₂×C₈×C₂₉	—	2	2	no	—
75	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
83	C₄⋊C₄×C₂₉	—	2	4	no	—
84	C₄⋊C₄×C₂₉	—	2	4	no	—
85	C₄⋊C₄×C₂₉	—	4	4	no	—
86	C₄⋊C₄×C₂₉	—	4	4	no	—
87	C₄⋊C₄×C₂₉	—	4	4	no	—
88	C₄⋊C₄×C₂₉	C₂×D₄	1	2	yes	—
89	C₄⋊C₄×C₂₉	C₂×D₄	1	4	yes	—
90	C₄⋊C₄×C₂₉	C₂×D₄	1	2	yes	—
91	C₂²⋊C₄×C₂₉	—	2	2	no	—
92	C₂²⋊C₄×C₂₉	—	2	2	no	—
93	C₄²×C₂₉	—	2	2	no	—
94	C₄²×C₂₉	—	2	2	no	—
95	C₄²×C₂₉	—	2	1	no	—
96	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
97	#28823	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
99	Q₈×C₂₉	—	16	4	no	—
100	C₂×C₄×C₂₉	—	2	2	no	—
101	C₂×C₄×C₂₉	—	2	2	no	—
102	C₂×C₄×C₂₉	—	2	4	no	—
103	C₂×C₄×C₂₉	—	2	2	no	—
104	C₂×C₄×C₂₉	—	4	2	no	—
105	C₂×C₄×C₂₉	—	4	4	no	—
106	C₂×C₄×C₂₉	—	8	4	no	—
107	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
113	C₈×C₂₉	—	2	4	no	—
114	C₈×C₂₉	—	2	4	no	—
115	C₈×C₂₉	C₄⋊Q₈	1	2	yes	—
116	C₈×C₂₉	C₄⋊Q₈	1	2	yes	—
117	C₈×C₂₉	C₄⋊Q₈	1	2	yes	—
118	C₈×C₂₉	C₄⋊Q₈	1	2	yes	—
119	#3164	C₂×C₂₉	1	2	yes	—
120	#3163	C₂×C₂₉	1	2	yes	—
121	#2061	C₂×C₂₉	1	4	yes	—
122	#1952	C₂×C₂₉	1	2	yes	—
123	#1951	C₂×C₂₉	1	2	yes	—
124	#1846	C₂×C₂₉	1	1	yes	characteristic
125	#1772	C₂×C₂₉	1	1	yes	characteristic
126	#1772	C₂×C₂₉	1	1	yes	characteristic
127	C₂²×C₂₉	—	2	2	no	—
128	C₂²×C₂₉	#544	1	1	yes	characteristic
129	C₂²×C₂₉	D₄²	1	1	yes	characteristic
130	C₂²×C₂₉	Q₈²	1	1	yes	characteristic
131	C₄×C₂₉	—	4	2	no	—
132	C₄×C₂₉	—	4	2	no	—
133	C₄×C₂₉	—	4	4	no	—
134	C₄×C₂₉	—	4	2	no	—
135	C₄×C₂₉	—	8	4	no	—
136	C₄×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
137	C₄×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
138	C₄×C₂₉	D₄×Q₈	1	1	yes	characteristic
139	C₄×C₂₉	D₄×Q₈	1	1	yes	characteristic
140	#544	C₂²×C₂₉	1	4	yes	—
141	#501	—	2	4	no	—
142	#501	—	2	4	no	—
143	#500	—	2	2	no	—
144	#499	—	2	2	no	—
145	#499	—	2	2	no	—
146	#498	—	2	2	no	—
147	C₂×#108	C₂²×C₂₉	1	1	yes	characteristic
148	C₂×#108	C₂²×C₂₉	1	1	yes	characteristic
149	#428	C₂²×C₂₉	1	4	yes	—
150	#428	C₂²×C₂₉	1	4	yes	—
151	#427	C₂²×C₂₉	1	1	yes	characteristic
152	#427	C₂²×C₂₉	1	1	yes	characteristic
153	C₂×#107	C₂²×C₂₉	1	2	yes	—
154	C₂×#106	C₂²×C₂₉	1	2	yes	—
155	#405	C₂²×C₂₉	1	2	yes	—
156	#405	C₂²×C₂₉	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
158	#382	C₂²×C₂₉	1	2	yes	—
159	#343	C₂²×C₂₉	1	4	yes	—
160	#339	C₂²×C₂₉	1	2	yes	—
161	#339	C₂²×C₂₉	1	2	yes	—
162	C₂×C₂₉	—	2	1	no	—
163	C₂×C₂₉	—	2	1	no	—
164	C₂×C₂₉	#2551	1	1	yes	characteristic
165	C₂×C₂₉	#3290	1	2	yes	—
166	C₂×M₁₆	C₂³×C₂₉	1	2	yes	—
167	C₂×M₁₆	C₂³×C₂₉	1	2	yes	—
168	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
170	C₄⋊Q₈	—	2	4	no	—
171	C₄⋊Q₈	—	2	4	no	—
172	#125	—	2	4	no	—
173	#125	—	2	4	no	—
174	C₂²⋊Q₈	—	4	4	no	—
175	#117	C₂³×C₂₉	1	2	yes	—
176	#117	C₂³×C₂₉	1	2	yes	—
177	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
178	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
179	#108	Q₈×C₂₉	1	4	yes	complemented
180	#108	Q₈×C₂₉	1	4	yes	complemented
181	#107	—	2	2	no	—
182	#107	—	2	2	no	—
183	#107	—	2	4	no	—
184	#107	—	2	4	no	—
185	#107	D₄×C₂₉	1	2	yes	—
186	#107	D₄×C₂₉	1	2	yes	—
187	#106	—	2	2	no	—
188	#106	—	2	2	no	—
189	#106	—	2	4	no	—
190	#106	—	2	4	no	—
191	#106	D₄×C₂₉	1	1	yes	characteristic
192	#106	D₄×C₂₉	1	1	yes	characteristic
193	#106	D₄×C₂₉	1	2	yes	—
194	#97	—	2	2	no	—
195	#97	—	2	2	no	—
196	C₄×C₈	—	2	2	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	Q₈×C₂₉	1	4	yes	complemented
199	#95	—	4	2	no	—
200	C₂₉	#28823	1	1	yes	characteristic, direct product factor, Sylow
201	C₂×Q₈	—	4	4	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	C₂×Q₈×C₂₉	1	4	yes	—
208	M₁₆	C₂×Q₈×C₂₉	1	4	yes	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
213	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
215	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
216	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
217	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
218	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	C₂×D₄×C₂₉	1	2	yes	—
225	C₄⋊C₄	C₂×D₄×C₂₉	1	4	yes	—
226	C₄⋊C₄	C₂×D₄×C₂₉	1	2	yes	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	1	no	—
232	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
233	C₂³	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
234	Q₈	—	16	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
244	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
245	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
246	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
247	C₂×C₄	C₂₉×2₋¹⁺⁴	1	1	yes	characteristic
248	C₈	—	2	4	no	—
249	C₈	—	2	4	no	—
250	C₈	C₂₉×C₄⋊Q₈	1	2	yes	—
251	C₈	C₂₉×C₄⋊Q₈	1	2	yes	—
252	C₈	C₂₉×C₄⋊Q₈	1	2	yes	—
253	C₈	C₂₉×C₄⋊Q₈	1	2	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	C₂₉×#544	1	1	yes	characteristic
256	C₂²	D₄²×C₂₉	1	1	yes	characteristic
257	C₂²	Q₈²×C₂₉	1	1	yes	characteristic
258	C₄	—	4	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	4	no	—
261	C₄	—	4	2	no	—
262	C₄	—	8	4	no	—
263	C₄	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
264	C₄	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
265	C₄	D₄×Q₈×C₂₉	1	1	yes	characteristic
266	C₄	D₄×Q₈×C₂₉	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	C₂₉×#2551	1	1	yes	characteristic
270	C₂	C₂₉×#3290	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₉	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₉
prime radical	C₂³×C₂₉	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
44. C₂₉×#108	Q₈	semidirect
45. C₂₉×#108	Q₈	semidirect
63. C₄×C₈×C₂₉	Q₈	semidirect
97. #28823	C₂₉	direct (N × H)
179. #108	Q₈×C₂₉	semidirect
180. #108	Q₈×C₂₉	semidirect
198. C₄×C₈	Q₈×C₂₉	semidirect
200. C₂₉	#28823	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₉×#3164	2
C₂₉×#3163	2
C₂₉×#2061	2
C₂₉×#1952	2
C₂₉×#1951	2
C₂₉×#1846	2
C₂₉×#1772	2
#28823	29

Subgroups — C29×#28823 #20323723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#28823 #20323723